Performance of density functional theory in computing nonresonant vibrational (hyper)polarizabilities

Metalloborospherenes with a Stabilized Classical Fullerene-like Borospherene B40 Act as Nonlinear Optical Switches, Electron Reservoirs, Molecular Capacitors, and Molecular Reactors

Using a new method of η5-Li and η6-Mg atoms capping the faces of the classical fullerene-like borospherene Td B40, we theoretically predict an exohedral metalloborospherene Td Mg10Li12&B40 molecule. Remarkably, a newfangled endoexo cage isomerism is proposed. Further, embedding Mg atoms in the Td B40 cage forms endohedral derivatives. Due to the intramolecular pull-push electron transfer relay, these obtained molecules possess unequal multilayered and alternant spherical charge distribution. The outer is an excess electron layer, bringing a molecular nonlinear switch character and an electron reservoir behavior with strong electron-donating and -accepting abilities. The middle (Mg2+)10(Li+)12 and the outer layers together constitute an electric double layer, presenting the behavior of a molecular capacitor where the electronic charge-discharge process occurs in the outer excess electron layer. The inner part is an empty cage B4026- with a strong negative electric field. The valence electrons of the embedded Mg atoms are transformed into new excess electrons and added in the outer excess electron layer, also exhibiting the charging behavior of the molecular capacitor. Considering the chemical reaction in the inner cage, the embedded Mg atom is ionized, forming an Mg2+ cation and 2e under the strong negative electric field; meanwhile, 2e is powerfully pushed into the outer excess electron layer. This chemical process shows a generalized Coulomb explosion, and thus the exohedral metalloborospherene molecules with cage B4026- may act as molecular reactors. The new species mark the genesis of classical fullerene-like borospherene chemistry and stimulate their applications in molecular nonlinear optical and nanoelectronics.

A quantum chemical investigation of the second hyperpolarizability of p-nitroaniline

Recent measurements of the third harmonic scattering responses of molecules have given a new impetus for computing molecular second hyperpolarizabilities (γ) and for deducing structure-property relationships. This paper has employed a variety of wavefunction and density functional theory methods to evaluate the second hyperpolarizability of the p-nitroaniline prototypical push-pull π-conjugated molecule, addressing also numerical aspects, such as the selection of an integration grid and the impact of the order of differentiation vs the achievable accuracy by using the Romberg quadrature. The reliability of the different methods has been assessed by comparison to reference Coupled-Cluster Singles and Doubles with perturbative treatment of the Triples results. On the one hand, among wavefunction methods, the MP2 scheme offers the best accuracy/cost ratio for computing the static γ. On the other hand, using density functional theory, γ remains a challenging property to compute because all conventional, global hybrid or range-separated hybrid, exchange-correlation functionals underestimate static γ values by at least 15%. Even tuning the range-separating parameter to minimize the delocalization errors does not enable to improve the γ values. Nevertheless, the original double-hybrid B2-PLYP functional, which benefits from 27% of PT2 correlation and 53% Hartree-Fock exchange, provides accurate estimates of static γ values. Unfortunately, the best performing exchange-correlation functionals for γ are not necessarily reliable for the first hyperpolarizability, β, and vice versa. In fact, the β of p-nitroaniline (pNA) could be predicted, with a good accuracy, with several hybrid exchange-correlation functionals (including by tuning the range-separating parameter), but these systematically underestimate γ. As for γ, the MP2 wavefunction method remains the best compromise to evaluate the first hyperpolarizability of pNA at low computational cost.

Effect of intermolecular interaction of the charge-transfer complex between molecular "tweezers" and C60/C70 on second-order nonlinear optical properties

To enhance understanding of the correlation between the intermolecular interaction and second-order nonlinear optical (NLO) properties, we studied a "molecular tweezer" with two corannulene substituents linked by a tetrahydro[5]helicene imide, which enabled highly sensitive and selective complexation of C₆₀/C₇₀ through convex-concave π-π interactions. The geometric structure, molecular orbitals, intermolecular interactions, electron absorption spectra and second-order NLO properties of the charge-transfer (CT) complexes formed by molecular tweezers and C₆₀/C₇₀ were studied by density functional theory. Larger fullerenes helped to increase the intermolecular interaction and CT, thereby increasing the first hyperpolarizabilities of CT complexes. Embedding of lithium ions helped to enhance the electron-absorption ability of fullerenes, thereby increasing the intermolecular interaction and intermolecular CT and, thus, enhancing their first hyperpolarizability significantly. Our data indicated that, through structure adjustment (including increasing the volume of fullerene and embedding alkali metal ions), we could enhance intermolecular interactions and improve intermolecular CT significantly. These actions could improve the second-order NLO properties of CT complexes.

Effects of Atypical Hydrogen Bonds and π-π Interactions on Nonlinear Optical Properties: Dimers of Triangular Structures Based on Perylene, Naphthalene, and Pyromellitic Diimides

Nonlinear optical (NLO) materials have become important materials in the field of high-speed optical devices due to the changes in light absorption and refraction caused by the photoelectric field. Compounds tend to exist as aggregates rather than single molecules, so intermolecular interactions are crucial to the nature of aggregates. Therefore, to study the effects of intermolecular interactions on nonlinear optical properties, we use a dimer simplified model and adopt the methods of controlling variables, which are the different intermolecular interactions resulting from the different stacking patterns of dimers based on the same monomer structures (2PMDI-1NDI and 2NDI-1PDI). It is found that compared with dimers involving π-π interactions, dimers involving C-H···O interactions have shorter intermolecular distances, larger intermolecular interaction energies, and smaller highest occupied molecular orbital-lowest unoccupied molecular orbital (HOMO-LUMO) energy gaps. Moreover, the C-H···O interactions are more conducive to the intermolecular charge transfers and more beneficial for increasing the nonlinear optical response values of aggregates with respect to π-π interactions. This work provides an important basis for the influence of intermolecular interactions on nonlinear optical properties.

The organic co-crystals formed using naphthalenediimide-based triangular macrocycles and coronene: intermolecular charge transfers and nonlinear optical properties

Formation of organic co-crystals is an effective strategy to synthesize near infrared emission and nonlinear optical (NLO) materials, which often show "1 + 1 > 2" performance. Moreover, the crystallization process can be effectively regulated through supramolecular interactions; thus the properties of co-crystal materials can also be flexibly regulated. Here, in order to further understand the nature and formation mechanism of co-crystals from the perspective of theoretical research, we studied the structures, intermolecular interactions, absorption spectra, charge transfer (CT) characteristics and nonlinear optical (NLO) properties of the newly synthesized organic co-crystals formed between naphthalenediimide based triangles (NDI, acceptor) and coronene (COR, donor). According to the analysis of decomposition of intermolecular interaction energy, dispersion energy played a major role, so the co-crystal properties can be regulated by regulating the intermolecular dispersion energy. More importantly, the formation of co-crystals NDI-COR and NDI-2COR reduced the E_gap values with respect to those of their components. And there was significant intermolecular CT from COR to NDI and the degree of CT in NDI-COR was larger than that in NDI-2COR, so that the α_tot and γ_tot values of NDI-COR and NDI-2COR were significantly greater than those of their components. Thus, the NLO properties of organic co-crystals can be further improved by enhancing the electron-donating ability of the donor and the electron-withdrawing ability of the acceptor to enhance the degree of intermolecular interaction energy and CT.

Hyperpolarizabilities of Push-Pull Chromophores in Solution: Interplay between Electronic and Vibrational Contributions

Contemporary design of new organic non-linear optical (NLO) materials relies to a large extent on the understanding of molecular and electronic structure-property relationships revealed during the years by available computational approaches. The progress in theory-hand-in-hand with experiment-has enabled us to identify and analyze various physical aspects affecting the NLO responses, such as the environmental effects, molecular vibrations, frequency dispersion, and system dynamics. Although it is nowadays possible to reliably address these effects separately, the studies analyzing their mutual interplay are still very limited. Here, we employ density functional theory (DFT) methods in combination with an implicit solvent model to examine the solvent effects on the electronic and harmonic as well as anharmonic vibrational contributions to the static first hyperpolarizability of a series of push-pull α,ω-diphenylpolyene oligomers, which were experimentally shown to exhibit notable second-order NLO responses. We demonstrate that the magnitudes of both vibrational and electronic contributions being comparable in the gas phase significantly increase in solvents, and the enhancement can be, in some cases, as large as three- or even four-fold. The electrical and mechanical anharmonic contributions are not negligible but cancel each other out to a large extent. The computed dynamic solute NLO properties of the studied systems are shown to be in a fair agreement with those derived from experimentally measured electric-field-induced second-harmonic generation (EFISHG) signals. Our results substantiate the necessity to consider concomitantly both solvation and vibrational effects in modeling static NLO properties of solvated systems.

First-order hyperpolarizabilities of propellanes: elucidating structure-property relationships

Following recent experimental work demonstrating strong nonlinear optical properties, namely second harmonic generation of light, in crystals composed of 16,20-dinitro-(3,4,8,9)-dibenzo-2,7-dioxa-5,10-diaza[4.4.4]propellane molecules [A. Miniewicz, S. Bartkiewicz, E. Wojaczyńska, T. Galica, R. Zaleśny and R. Jakubas, J. Mater. Chem. C, 2019, 7, 1255-1262] in this paper we aim to investigate "structure-property" relationships for a series of 16 propellanes presenting a wide palette of substituents with varying electron-accepting/donating capabilities. To that end, we use electronic- and vibrational-structure theories and a recently developed generalized few-state model combined with a range-separated CAM-B3LYP functional to analyze electronic and vibrational contributions to the first hyperpolarizability for the whole series of molecules. The variations in computed properties are large among the studied set of substituents and can reach an order of magnitude. It has been demonstrated that the maximum values of frequency-independent first hyperpolarizability are expected for strong electron-accepting NO₂ substituents, but only at the preferred position with respect to the electronegative oxygen atom in the 1,4-oxazine moiety. This holds for electronic as well as vibrational counterparts.

A generalized few-state model for the first hyperpolarizability

The properties of molecules depend on their chemical structure, and thus, structure-property relations help design molecules with desired properties. Few-state models are often used to interpret experimental observations of non-linear optical properties. Not only the magnitude but also the relative orientation of the transition dipole moment vectors is needed for few-state models of the non-linear optical properties. The effect of the relative orientation of the transition dipole moment vectors is called dipole alignment, and this effect has previously been studied for multiphoton absorption properties. However, so far, no such studies are reported for the first hyperpolarizability. Here, we present a generalized few-state model for the static and dynamic first hyperpolarizability β, accounting for the effect of dipole alignment. The formulas derived in this work are general in the sense that they can be used for any few-state model, i.e., a two-state model, a three-state model, or, in general, an n-state model. Based on the formulas, we formulate minimization and maximization criteria for the alignment of transition dipole moment vectors. We demonstrate the importance of dipole alignment by applying the formulas to the static first hyperpolarizability of ortho-, meta-, and para-nitroaniline. The formulas and the analysis provide new ways to understand the structure-property relationship for β and can hence be used to fine-tune the magnitude of β in a molecule.

A new tuned range-separated density functional for the accurate calculation of second hyperpolarizabilities

Among the nine functionals benchmarked, the most accurate γ are obtained by Tα-LC-BLYP, reducing about half the errors of LC-BLYP.

DFT and spatial confinement: a benchmark study on the structural and electrical properties of hydrogen bonded complexes

An extended set of 37 exchange correlation functionals, representing different DFT approximations, has been evaluated on a difficult playground represented by the dipole moment (μ_z), polarizability (α_zz), first hyperpolarizability (β_zzz), and the corresponding interaction-induced electrical properties (Δμ_z, Δα_zz, Δβ_zzz) of spatially confined hydrogen bonded (HB) dimers. A two-dimensional harmonic oscillator potential was used to exert the effect of spatial restriction. The performance of DFT methods in predicting hydrogen bond lengths in the studied molecular complexes upon confinement has also been examined. The data determined using a high-level CCSD(T) method serve as a reference. The conducted analyses allow us to conclude that methods rooted in DFT constitute a precise tool for the calculation of μ_z and α_zz as well as Δμ_z and Δα_zz, as most of the tested functionals provide results affected by rather small relative errors. On the other hand, an accurate description of the nonlinear optical response of the studied HB systems remains a great challenge for most of the analyzed DFT functionals, both in vacuum and in the presence of an analytical confining potential. Some of the tested DFT methods are found to be prone to catastrophic failure in the prediction of β_zzz as well as Δβ_zzz. The obtained results indicate that there is no great chasm in performance between functionals belonging to different DFT approximations or functionals including different amount of Hartree-Fock exchange when the values of dipole moment and first hyperpolarizability as well as the corresponding interaction-induced electrical properties are considered. However, a higher fraction of Hartree-Fock exchange improves the quality of predictions of α_zz and Δα_zz. Additionally, it has been shown that only three functionals from the examined set, namely B2PLYP, B3LYP and ωB97X-D, provide highly accurate structural parameters for the investigated systems. Of significant importance is the conclusion that the ωB97X-D functional, representing a modern and highly parametrized range-separated hybrid, demonstrates the most coherent behavior, showing rather small deviations from the reference data in the case of μ_z, α_zz, Δμ_z and Δα_zz as well as the structural parameters of the studied HB dimers. Moreover, our results indicate that the presence of spatial confinement has a rather small effect on the performance of DFT methods.

Can Density Functional Theory Be Trusted for High-Order Electric Properties? The Case of Hydrogen-Bonded Complexes

This work reports on an extensive assessment of the performance of a wide palette of density functional approximations in predicting the (high-order) electric properties of hydrogen-bonded complexes. To this end, we compute the electronic and vibrational contributions to the electric polarizability and the first and second hyperpolarizabilities, using the CCSD(T)/aug-cc-pVTZ level of theory as reference. For all the studied properties, the average absolute errors below 20% can only be obtained using the CAM-B3LYP functional, while LC-BLYP and MN15 are shown to be only slightly less accurate (average absolute errors not exceeding 30%). Among Minnesota density functionals, i.e., M06, M06-2X, and MN15, we only recommend the latter one, which quite accurately predicts the electronic and vibrational (hyper)polarizabilities. We also analyze the optimal tuning of the range-separation parameter μ for the LC-BLYP functional, finding that this approach does not bring any systematic improvement in the predictions of electronic and vibrational (hyper)polarizabilities and the accuracy of computed properties is largely system-dependent. Finally, we report huge errors in predicting the vibrational second hyperpolarizability by ωB97X, M06, and M06-2X functionals. Based on the explicit evaluation of anharmonic terms contributing to the second hyperpolarizability, this failure is traced down to a poor determination of third- and fourth-order energy derivatives with respect to normal modes. These results reveal serious flaws of some density functional approximations and suggest caution in selecting the appropriate functional to calculate not only electronic and vibrational (hyper)polarizabilities but also other molecular properties that contain vibrational anharmonic contributions.

Partition of optical properties into orbital contributions

A new tool to analyze the response property through the partition of nonlinear optical properties in terms of orbital contributions (PNOC), valuable in the assessment of the electronic structure methods in the NLOPs computations, is presented.

Unusual response to environmental polarity in a nonlinear-optical benzylidene-type chromophore containing a 1,3-bis(dicyanomethylidene)indane acceptor fragment

One of the strongest known electron-accepting fragments used in the synthesis of organic dyes for applications in nonlinear optics (NLO) is 1,3-bis(dicyanomethylidene)indane (BDMI). By studying a benzylidene-type push-pull chromophore bearing a 5-carboxy-BDMI electron-acceptor and 4-(dimethylamino)aniline donor fragment, we demonstrate that this class of compounds can show unusual response to the polarity of the surrounding medium. The combined results of UV-Vis absorption spectrometry, NMR experiments and computational modeling indicate that the studied compound undergoes a geometrical transformation that involves an increase in the torsion angle ω between the aniline and indane ring systems with the rise of the polarity of the surrounding medium. This process is partly facilitated by an increased rotational freedom around ω in more polar solvents, as detected experimentally by NMR and predicted by calculations. Regarding the practical application aspects, computations predict that the solvent-polarity-induced increase of torsion ω would lead to a notable decrease in the first hyperpolarizability (β) value. This was detected experimentally, as hyper-Rayleigh scattering (HRS) data showed a drop in the compound's NLO activity from β_HRS(532)  = 513 × 10^-30 esu in toluene to β_HRS(532)  =  249 × 10^-30 esu in acetonitrile. This places limitations on the NLO applications of the studied compound and its structural analogues, as the surrounding medium (solvent of polymer matrix) with the lowest possible polarity needs to be used to maximize their NLO efficiency.

The novel link between planar möbius aromatic and third order nonlinear optical properties of metal-bridged polycyclic complexes

Metal–bridged polcyclic aromatic complexes, exhibiting unusual optical effects such as near-infrared photoluminescence with particularly large Stokes shifts, long lifetimes and aggregation enhancement, have been established as unique “carbonloong chemistry”. Herein, the electronic structures, aromaticities, absorption spectra and third order nonlinear optical (NLO) responses of metal–bridged polcyclic aromatic complexes (M = Fe, Re, Os and Ir) are investigated using the density functional theory computations. It is found that the bridge–head metal can stabilize and influence rings, thus creating π–, σ– and metalla–aromaticity in an extended, π–conjugated framework. Interestingly, metal radius greatly influence the bond, aromaticity, liner and third order NLO properties, which reveals useful information to develop new applications of metal regulatory mechanism in NLO materials field. Significantly, the novel relationship between the aromaticity and third order NLO response has firstly been proposed, that the metal-bridged polycyclic complex with larger aromaticity will exhibit larger third order nonlinear optical response. It is our expectation that the novel link between aromaticity and NLO response could provide valuable information for scientists to develop the potential NLO materials on the basis of metal–bridged polycyclic complexes.

The behavior of a paramagnetic system in electric and magnetic fields as exemplified by revisiting Li@B10H14

The electric and magnetic properties of the Li@B₁₀H₁₄ complex, considered as a novel inorganic electride-type system with potential for second-order non-linear optical (NLO) applications, have already been studied. However, the reported C_2v structure is not the global energy minimum and therefore its electronic and magnetic properties need to be revisited. Moreover, by applying more accurate computational protocols (ROHF-CCSD/CCSD(T) and larger basis sets) we show that the model chemistry used earlier (UMP2/6-31+G(d)) is not sufficient for reliable description of the NLO responses of this open-shell doublet complex. The global minimum based on the C_s symmetry is significantly (by ca. 25 kcal mol^-1) more stable than the C_2v structure and it should be viewed as a system with moderate NLO responses. An excess of unpaired electron density is also responsible for the contact and pseudo-contact contributions to the magnetic properties, which was not considered in the earlier work.

On the physical origins of interaction-induced vibrational (hyper)polarizabilities

This paper presents the results of a pioneering exploration of the physical origins of vibrational contributions to the interaction-induced electric properties of molecular complexes. In order to analyze the excess nuclear relaxation (hyper)polarizabilities, a new scheme was proposed which relies on the computationally efficient Bishop-Hasan-Kirtman method for determining the nuclear relaxation contributions to electric properties. The extension presented herein is general and can be used with any interaction-energy partitioning method. As an example, in this study we employed the variational-perturbational interaction-energy decomposition scheme (at the MP2/aug-cc-pVQZ level) and the extended transition state method by employing three exchange-correlation functionals (BLYP, LC-BLYP, and LC-BLYP-dDsC) to study the excess properties of the HCN dimer. It was observed that the first-order electrostatic contribution to the excess nuclear relaxation polarizability cancels with the negative exchange repulsion term out to a large extent, resulting in a positive value of Δα(nr) due to the contributions from the delocalization and the dispersion terms. In the case of the excess nuclear relaxation first hyperpolarizability, the pattern of interaction contributions is very similar to that for Δα(nr), both in terms of their sign as well as relative magnitude. Finally, our results show that the LC-BLYP and LC-BLYP-dDsC functionals, which yield smaller values of the orbital relaxation term than BLYP, are more successful in predicting excess properties.

Vibrational corrections to the second hyperpolarizabilities of Al(n)P(n) clusters

In this work, we report results of vibrational corrections to the second hyperpolarizabilities of Al2P2, Al3P3, Al4P4, Al6P6, and Al9P9 clusters. The vibrational corrections were calculated through the perturbation theoretic method of Bishop and Kirtman and also using a variational methodology at the second order Møller-Plesset perturbation theory level with the aug-cc-pVDZ basis set. Results show that the vibrational corrections are important, accounting for more than half of the corresponding electronic second hyperpolarizabilities at the static limit. Comparisons between results obtained through both methods show very good agreements for the terms [α(2)] and [μβ] but significant differences for the term [μ(2)α]. Dynamic vibrational corrections to the second hyperpolarizabilities related to the dc-second harmonic generation, intensity dependent refractive index, and dc-Kerr nonlinear optical processes are also reported.

The first hyperpolarizability of nitrobenzene in benzene solutions: investigation of the effects of electron correlation within the sequential QM/MM approach

Molecular polarization due to solvation amplifies the impact of electron correlation on the first hyperpolarizability of nitrobenzene.

Can short- and middle-range hybrids describe the hyperpolarizabilities of long-range charge-transfer compounds?

The hyperpolarizabilities of five prototypical and four recently synthesized long-range charge-transfer (CT) organic compounds are calculated using short- and middle-range (SR and MR) hybrid functionals. These results are compared with data from MP2 and other DFT methods including GGAs, global hybrids, long-range corrected functionals (LC-DFT), and optimally tuned LC-DFT. Although it is commonly believed that the overestimation of hyperpolarizabilities associated with CT excitations by GGA and global hybrid functionals is the result of their wrong asymptotic exchange potential, and that LC-DFT heals this issue, we show here that SR and MR functionals yield results similar to those from LC-DFT. Hence, the long-range correction per se does not appear to be the key element in the well-known improved description of hyperpolarizabilities by LC-DFT. Rather, we argue that the inclusion of substantial amounts of Hartree-Fock exchange, which reduces the many-electron self-interaction error, is responsible for the relatively good results afforded by range separated hybrids. Additionally, we evaluate the effects of solvent and frequency on hyperpolarizabilities computed by SR and MR hybrids and compare these predictions with other DFT methods and available experimental data.

Stereo-electronic, vibrational, and environmental contributions to polarizabilities of large molecular systems: a feasible anharmonic protocol

Reliable computations of linear and non-linear optical properties of molecular systems in condensed phases require a proper account of stereo-electronic, vibrational, and environmental effects. In the framework of density functional theory, these effects can be accurately introduced using second-order vibrational perturbation theory in conjunction with polarizable continuum models. We illustrate the combination of an anharmonic description of the ground-state potential energy surface with solvation effects treated with the polarizable continuum model (PCM) in the calculation of the electronic, zero-point, and pure vibrational polarizabilities of selected systems. The description of the solvation environment is enriched by taking into account the dynamical aspects of the solute-solvent interactions through the inclusion of both electronic and vibrational non-equilbrium effects, as well as the direct effect of the solvent on the electric field that generates the molecular response (local field effect). This treatment yields accurate results which can be directly compared with experimental findings without the need of empirical corrections.

Electronic responses of long chains to electrostatic fields: Hartree-Fock vs. density-functional theory: a model study

The response to an electrostatic field is determined through simple model calculations, within both the restricted Hartree-Fock and density functional theory methods, for long, finite as well as infinite, periodic chains. The permanent dipole moment, μ0, the polarizability, α, and the hyperpolarizabilities β and γ, calculated using a finite-field approach, are extensively analyzed. Our simple model allows for treatment of large systems and for separation of the properties into atomic and unit-cell contributions. That part of the response properties attributable to the terminations of the finite system change into delocalized current contributions in the corresponding infinite periodic system. Special emphasis is placed on analyzing the reasons behind the dramatic overestimation of the response properties found with density functional theory methods presently in common use.

Analytic density functional theory calculations of pure vibrational hyperpolarizabilities: the first dipole hyperpolarizability of retinal and related molecules

We present a general approach for the analytic calculation of pure vibrational contributions to the molecular (hyper)polarizabilities at the density functional level of theory. The analytic approach allows us to study large molecules, and we apply the new code to the study of the first dipole hyperpolarizabilities of retinal and related molecules. We investigate the importance of electron correlation as described by the B3LYP exchange-correlation functional on the pure vibrational and electronic hyperpolarizabilities and compare the computed hyperpolarizabilities with available experimental data. The effects of electron correlation on the pure vibrational corrections vary signficantly even between these structurally very similar molecules, making it difficult to estimate these effects without explicit calculations at the density functional theory level. As expected, the frequency-dependent first hyperpolarizability, which determines the experimentally observed second-harmonic generation, is dominated by the electronic term, whereas for the static hyperpolarizability, the vibrational contribution is equally important. As a consequence, frequency extrapolation of the measured optical hyperpolarizabilities can only provide an estimate for the electronic contribution to the static hyperpolarizability, not its total value. The relative values of the hyperpolarizabilities for different molecules, obtained from the calculations, are in reasonable agreement with experimental data.